# Winter 2019

In Winter 2019, important dates and times for WDRP are as follows:

**Wednesday Jan 9th, time 5:00 pm: **Start-of-quarter kick-off event in ECE 045.

**Wednesday Mar 13th, time 5:00 pm: **End-of-quarter presentations in SAV 140 and 141.

Outside of these times, undergraduate students and graduate students are expected to meet for 1 hour per week at a time of their choosing. In between meetings, undergraduate students are also expected to spend 3-4 hours per week reading and working on their own. Graduate students are expected to take the requisite time prepare for meetings on their own time as well. To facilitate meeting we have ECE 045 reserved Mondays and Wednesdays from 5:00-6:20 pm.

Winter 2019 Projects

**What is a number?
**

*Mentor: Lucas Van Meter*

*Mentee: Trayana Rogers*

What exactly counts as a number anyways? (pun intended) How do mathematicians start with whole numbers and end up with complex numbers or infinite numbers? We will investigate various types of numbers possibly including whole, fractions, negative, algebraic, transcendental, infinitesimal and transfinite, complex and quaternion. The primary text will be “The Book of Numbers” by Conway and Guy.

**Integer Partitions
**

*Mentor: Stark Ledbetter*

Mentee: Kenny Le

Mentee: Kenny Le

In addition to posing one of the most elementary unsolved problems in mathematics, the theory of integer partitions has fostered many results, including Euler’s identity and the Rogers-Ramanujan Identities. We will explore these and more through Andrews and Eriksson’s book, Integer Partitions.

**Random Walks
**

*Mentor: Peter Lin*

Mentee: Jamie Forschmiedt

Mentee: Jamie Forschmiedt

Many random processes in science, engineering, finance, and math can be modeled as a ‘random walk’. Even though the random walk is random, there are many interesting things we can say about it. We could try to understand some of these things, such as Polya’s theorem on random walks. References: Random walks and electric networks, by Doyle and Snell Understanding Probability: Chance rules in everyday life, by Henk Tijms

**Optimal Mass Transport
**

*Mentor: Mathias Hudoba de Badyn*

*Mentee: Shannon Mallory*

Optimal mass transport is the rigorous study of moving piles of dirt (probability measures) with the least amount of effort (cost functions). It’s used in everything from training neural networks to controlling swarms of robots. In this project, we will go over the basics of optimization building up to the formulation of OMT, and then explore both theory and applications based on student interest.

**The Banach-Tarski Paradox
**

*Mentor: Sean McCurdy*

*Mentee: Jon Kim*

The Banach-Tarski paradox is a construction by which a ball is cut up into finitely many pieces, and then these pieces are reassemble into two balls of the same size as you started with. Sounds impossible, right? This project would explore notions of the “size” of a set, introduce “paradoxical” sets, and some elementary group theory to construct these “paradoxical” decompositions. After introductory material, the book by Wagoner will be a guide.

**Understanding Markov Chains
**

*Mentor: Anthony Sanchez*

*Mentee: Steven Nguyen*

The aim of this project would be to learn about Markov chains. Markov chains are a random process that arise in many theoretical contexts and real world situations. We will read out of the friendly text “Understanding Markov Chains”. As such, this directed reading would be perfect for any student who has some background in probability (Math 394 would be ideal), linear algebra (Math 308 ), and wishes to further their knowledge in probability theory.

**Graph Theory
**

*Mentor: Catherine Babecki*

*Mentee: Milo Nguyen*

Learn about graph theory! In the broadest strokes, a graph is a collection points and some connections between them. Graphs can be used to model migration of animal species, represent neurological connections in the brain, analyze the origins of families of languages, and determine who Facebook should put on your recommended friends list. We will most likely use the book “Introduction to Graph Theory” by Douglas West, but could use a more advanced textbook depending on the level of the student.

**Beyond Linear Algebra
**

*Mentor: Alessandro Slamitz*

*Mentee: Griffin Withington*

We’ll resume from the topics of MATH 308 and dig more into linear algebra. We can either decide to keep this project purely theoretical or seeing applications and how to implement some algorithms. If we choose the theoretical paths we will need other important concepts in mathematics such as groups, rings, fields and topological spaces. If we decide to work on applications we will need to use softwares such as MATLAB.

**Young Tableaux
**

*Mentor: Peter Gylys-Colwell*

*Mentee: Jack Sundsten*

Young Tableaux are fascinating combinatorial objects to pop up in main stream mathematical interest within the past century. Their use has lead to powerful classifications in representation theory and algebraic geometry. In this reading course we will explore what a Young Tableau is and the combinatorics developed in counting and classifying them.

**Representation theory
**

*Mentor: Graham Gordon*

*Mentee: Thalya Paleologu*

Like linear algebra? Like permutations? You might like representation theory! It is a way to study abstract algebraic structures called “groups” using only tools from linear algebra. Bruce Sagan’s book, “The Symmetric Group,” introduces representation theory in general and then discusses how it applies to the structure of the symmetric group. We could try reading some of this book, computing some examples, and doing some math together.

**Galois theory
**

*Mentor: Tuomas Tajakka*

*Mentee: Rohan Hiatt*

Have you ever wondered why polynomials of degree 2, 3, and 4 can be solved using a formula, but there’s no such formula for degree 5 polynomials? The answer is given by Galois theory, the study of field extensions and their symmetries. Galois theory is part of the basic toolkit of a modern number theorist or algebraic geometer, and as another basic application, one can prove the impossibility of some classical compass-and-straightedge constructions. Possible study material could be Galois Theory by Ian Stewart, or Field and Galois Theory by James Milne.

**Arithmetic Geometry and Number theory
**

*Mentor: Sam Roven*

*Mentee: Blanca Viña*

We will explore various sub fields of number theory, arithmetic/algebraic geometry, and algebraic number theory.